One of my favorite aspects of math is how, in many cases, we developed new tools to account for the limitations and inefficiencies of the old ones. One of my favorite parts of math education is how we can highlight those inefficiencies with a well-posed problem and justify the use of new tools.

For instance, let’s say some students are awesome at calculating the slope between (-2, 5) and (1, 11) by counting unit squares by hand and then dividing. (ie. “Three over, six up. The slope is two.”) Unhelpful High School Teacher teaches them the slope formula, assigns them the same kind of problems, and punishes the students who don’t use the newer tool even though the older tool is easier. Helpful High School Teacher asks the students to find the slope between (-2, 5) and (999998, 2000005).

“Helpful High School Teacher asks the students to find the slope between (-2, 5) and (999998, 2000005).”

Such a simple/evil way to show the importance of both methods!

I’m a first year math teacher who, despite impressing colleagues with my work ethic, really found myself treading water and relying on awful lectures at the end of my first half year of teacher (also, coaching two sports hasn’t helped :/).

I’ve been parsing the many contents of this site and have spent the majority of my break adopting two major things: relying on slides (PPT in my case) and your assessment method. I’m going to implement the assessment method in my Pre-Calc classes immediately (a 50% average on our last test made this one an easy decision).

For my Geometry class I’ve been been able to pick and choose some great stuff here to integrate in and help me avoid falling back on lectures. Since this is my first year of Pre-Calc – student teaching included – however, I’ve found this task much more difficult with that class.

Thanks again for the great content! And if anyone can steer me in the direction of some multimedia stuff about trig/pre-calc, that would be great too :)

@Tom, for “multimedia stuff on trig/pre-calc” I can offer no better advice than this: learn to use Geogebra really well. If you google around I’m sure you’ll find some files ready to use, but the most important thing is to be able to create files that support the way you want to teach the concepts. That assumes you have a projector in your room and/or students have access to computers.

Underscores the need to emphasize multiple methods to students…I explain this to students by drawing a “Begin Problem” box with several different arrows emanating from it and eventually arriving at a “Finish Problem (aka “Solution”) box. Some are long, some short, some corkscrew, but all end up at the same spot. I’ve used this model with my algebra 1 students and AP calc students and it has helped them all get it, perhaps the AP calc students the most since many have been so co-opted into thinking there is only one “right” way to work a problem.

What about this? Determine the slope using two/three different methods. This validates each methods. I try to arm my students with as many methods as possible and usually tell them to use the method they think will work best under any given situation.

It seems to me that the whole class would benefit from seeing how both (or more) of the answers are connected. This would certainly ease concerns from people like me that some students are simply following the algorithm of finding slope using a formula and not really making the big connection.

Also, I have found that students who only rely on the formula rarely have a solid conceptual understanding of positive versus negative slope.

@Jim — I know the feeling. I’ve spent a fair amount of time having my students figure out the slope formula by counting blocks, increasing the number of blocks to count, then eventually projecting a huge grid of teeny tiny little blocks that they could tell were blocks, but had a hard time counting because a) I wouldn’t let them out of their seats to the board to see better and b) because by the time you get to the 47th block, you probably lose count. Anyway, after all the time it took one of them to say something like “just subtract the y values and the x-values,” I had some students say, even if under their breath so I couldn’t hear…I wish he would have just showed us that in the first place, this was a waste of time. I think my head was in the right place, but after being 9 years of having math be, as you say, “knowing which formula to use and remembering that formula,” is it right of me to try to un-teach that mentality and preach understanding versus memorization? I think I just wanted to join in, too…

And I suspect a good answer to the question I posed would be yes, it is! But maybe I phrased it incorrectly. Can it be done in the amount of time that I have in a year?

Curmudgeonly Math Teacher asks for the slope between
(-2, 5) and (99999998, 2000000005) without using commas or spacing and smiles that evil grin that tells the students that a curve ball just went whizzing by:
(-2, 5) and (99 999 998, 2 000 000 005)

I’ve done a couple of applications with rates of change in the past that I’m pretty proud of and students get a lot of buy-in from:
1. Flow rates of fire hydrants–this banks on my experiences from having spent years on our local volunteer fire department and the fact our community plays by the NFPA 291 standard pertaining to hydrant cap colors (available at http://firehydrant.org/info/hycolor.html if you want to try it yourself). I inform students that the fire engine arrives on scene with 1000 gallons in its tank, which surprises some students who think it arrives on scene empty, with just tools. Once the truck is connected to a hydrant water supply, and knowing its cap color, we can find the amount of time for the truck to dispense 10,000 or x number of gallons of water on the scene, or note that after 8 minutes there has been 10,000 gallons pushed through the truck’s pump. Either way gives them an extension of the topic at hand and gets students talking about how to account for the amount on the truck, what the output rate was from the truck, what 200 psi feels like on the end of a hose nozzle, if hydrant pressure would be considered the input or output variable in this scenario, and things like that. REAL WORLD STUFF! I especially like students taking note of the hydrant nearest their home (again, the firefighter in me appreciates the awareness) and what factors might play a role in why one student’s hydrant color differs from someone else’s, such as proximity to a water tower, the age of infrastructure in their part of town, or surrounding hills that might diminish water pressure. There is little, if any, cost behind this lesson and the link I provided earlier gives insight into the flow rates used.

2. Elevation change at access ramps–I became fascinated with all the accomodations made in building construction by the ADA (Americans with Disabilities Act) in college and used ramp slope as part of a lesson when I student taught (ramp steepness is not to exceed 10%). When I first did the lesson, we happened upon a ramp that was not up to code, which led my students to sprint to the principal’s office, singing about how school should shut down because it’s in violation and such. Principal appreciated their enthusiasm and that they grasped the concept, but asked me to find a different application at the current time. Now, a handful-and-a-half years later, I was lucky enough to use GPS devices to show students the elevation at a certain point, mark a POI (“point of interest” in GPS-speak) and measure the distance to that location. This gave us a real world “rise” and “run” behind the topic. Students liked this one as well and, now at a different school with different principal, I’ve had administrators jump in on the actvity and ask the students to explain it to them. A fun side note is how students talk about positive and negative slope in context of which way you’re going, as well as extensions into zero (an ordinary hallway or the football field) and undefined slopes (the flag pole).

Sorry this was so long, but a couple previous commenters hinted that they were looking for applications for such a topic.

The best motivations are the ones where you show the students the limitations of the method they’re currently using. I always use the, ‘oh, you think you’re so smart … well try this’. And funny enough a couple weeks back when I did the lessons on slope (and yes, I used your motivation with the picture of the ski lift and we had that great conversation that motivated the need to label the points and the need for the coordinate plane) (and yes, I did say lessonS because the first day we spent on counting boxes and the next day we talked about the formula), I gave them a similar problem where they had to find the slope of two ridiculous coordinates. Students were able to extrapolate from the smaller points to get the formula.

“This year one of my seniors in Honors calculus told me that she always thought that math was about knowing which formula to use and remembering that formula. I wanted to weep but this was not her fault, it was the fault of how she has been assessed and rewarded along the way.”

I do not believe that teachers or schools are responsible for this mindset. This mindset seems to me to be as old as humans, and if anything, teachers simply give in to it, they don’t create it. I think it boils down to the simple fact that some students are interested enough in a subject to want to know how it works, while (most) others are not.

I think that is why assessments are what they are. Any one of us could easily devise assessments that would fail the “formula” student (and most of the class). Some students run with this stuff while many others can barely even walk with it, and when you mandate that students take these classes then you are eventually going to either compromise your ideals, or fail most of the class.

I like your addition of step #2 “The teacher teaches the general expression for slope, but sticking to the familiar concrete examples so that the students “trust” the new method(s).”

As for intuition in students, I feel it needs to be explicitly taught to students rather than assumed, and in my own past experiences, my best teachers have tried to do that. An excellent calculus professor I once had would constantly be saying things like “…and then when you’re finished, look at the answer. Does it look right? If you get a number that’s not between 4 and 6, there’s obviously something wrong, isn’t there?”; or “…and if you don’t believe me, or you get on the exam and your mind completely blanks, try out a few easy examples and work it out. Convince yourself it’s right!”

I think for the best math students, a lot of this advice is what they’re doing anyways whether they realize it or not, but it’s not obvious for the average students.

Bizarre that I was thinking how much I despise the slope formula a few days before reading this post. My frustration is that I find many students unable to find the slope if they forget the formula. Give them a line with points at (-2, 5) and (1, 11) and take away their slope formula and they gaze at you like you have just asked them to prove Fermat’s Last Theorem. “Give me back my slope formula!” I understand it has a place and purpose (i.e. (-2, 5) and (999998, 2000005)). However, I think teacher’s are missing it if they aren’t making students proficient in both the “old” and “new” method. Thanks for the sharpening.

Christine said
“We need to give students (and teachers) a wonderful mathematical toolkit, the confidence and knowledge of which tools to use and if when applied, whether or not a correct answer is produced.”

While I agree with the first part (about treating mathematical skills as a toolkit), as an engineering professor, it is important to me that the correct answer be produced as well. Getting the right answer by magic or cheating is useless, but getting the wrong answer (even with “understanding”) is also useless.

I’m late to this commenting party but FWIW I experienced this same phenomenon myself just before break with a colleague at school.

I was consulting with her during a class period on a project, and decided to offer some assistance checking student work on the side. I was using my tested and true plong multiplication method to check student’s work when the teacher I was consulting with looked at with almost a bemused and bewildered look on her face.

“You don’t use the partial products method?” was her shocked question at assessing my math skills.

I felt a bit ashamed that A.) I had never learned the partial products method, and that B.) I wasn’t doing it the way she did. Don’t get me wrong, it took about 15 seconds for that sense of shame to go away before confidently telling her as long as I got the answer, it didn’t concern me. I’d accomodate her students, but how I do things is just fine with me :)

A problem could also take the shape of a common problem in engineering or some other field:

An two resistors are connected in series, one resistor is known and has a resistance value of 1 ohm. The second resistor is unknown. Using a voltmeter the voltage drop across the two resistors is measured. The voltage drop across the 1 ohm resistor is 1 V and across two unknown resistor is 100 V.

a) Calculate the resistance of the unknown resistor using ohm’s law Voltage=Current*Resistance.

b) Calculate the resistance taking into account that the voltmeter has an accuracy of 1 %.

in this case counting squares can be used at (a) and the formula in (b).

I’m not sure I understand this. Doesn’t one teach about this by suggesting that slope is the common notion of how steep something is?

Why would someone need to count blocks? The key notion is that we measure slope with a number as rise/run. Do you really have to have kids count blocks to know the rise and run? Can they not see that this is a difference? Boy, if that’s the problem, we really are in trouble as kids should be doing this stuff long before we get to a discussion of slope, no?

As for the example:

“Helpful High School Teacher asks the students to find the slope between (-2, 5) and (999998, 2000005).”

Rather than see this as some confirmation that a formula is useful, it struck me as a great opportunity to talk about using estimation to get a reasonably accurate result without a pencil and paper.

Andy, I agree with your comments. Formulas and calculators are great, but being able to estimate in your head, understand significant digits, know what is a reasonable answer, etc. are important skills to develop and use in life.